1 Executive Summary

Purpose of this report:

To profile who performs the best and worst at junior-level mathematics at the University of Sydney.

Stakeholder:

Industry corporations looking to attract intelligent junior mathematicians from the University of Sydney for internships or employment.

Main discoveries:

The best junior mathematicians are typically - full time domestic students aged 18 and under
In contrast, the worst performing junior mathematicians are generally - part time international men aged over 25
There are no clear distinctions in gender and the semester of study between the best and the worst performing students

2 Initial Data Analysis (IDA)

2.1 A Glimpse of the Data Set

# Importing the data set from local .csv file
data = read.csv("Data.csv")

# Changing the variable names into snake case
data = clean_names(data)

# Quick look at the data set
datatable(data, rownames = F, filter = "top", options = list(pageLength = 5, scrollX = T))

2.2 Assessing R’s Classication of the Variables

# Size of the data and R's classification of the variables
str(data)

## 'data.frame':    10845 obs. of  23 variables:
##  $ student_enrolment_key           : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ unit_of_study_identifier        : Factor w/ 15 levels "UNIT A","UNIT C",..: 4 9 9 9 5 9 3 11 7 12 ...
##  $ semester                        : int  2 1 1 2 1 1 2 2 1 2 ...
##  $ mode_of_study                   : Factor w/ 3 levels "Full Time","Part Time",..: 1 1 1 1 2 1 1 1 1 1 ...
##  $ gender                          : Factor w/ 2 levels "F","M": 1 2 2 1 2 2 1 2 1 2 ...
##  $ age_category                    : Factor w/ 4 levels "18 and Under",..: 2 2 2 3 3 2 1 2 2 1 ...
##  $ dom_int                         : Factor w/ 2 levels "Domestic","International": 2 2 2 2 1 1 1 2 1 1 ...
##  $ mark                            : int  68 41 42 46 60 75 82 79 67 62 ...
##  $ canvas_access_week_1            : int  1 1 1 0 1 1 1 1 1 1 ...
##  $ canvas_access_week_2            : int  1 1 1 0 1 1 1 1 1 1 ...
##  $ canvas_access_week_3            : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_4            : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_5            : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_6            : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_7            : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_8            : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_9            : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_10           : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_11           : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_12           : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_week_13           : int  1 1 0 1 1 1 1 1 1 1 ...
##  $ canvas_access_mid_semester_break: int  1 1 1 1 1 1 1 1 1 1 ...
##  $ canvas_access_stuvac            : int  1 1 1 1 1 1 1 1 1 1 ...

The semester variable should be changed from integer to a factor because it takes only two possible values (1 and 2).

In addition, the class of each canvas_access_... variable should be logical rather than an integer, where 1 is TRUE and 0 is FALSE.

# Changing the class of `semester` from integer to factor
data$semester = as.factor(data$semester)

# Changing the class of each `canvas_access_...` variable from integer to logical
for(i in c(9:(ncol(data) - 1))) {
    data[,i] = as.logical(data[,i])
}

2.3 Source of the Data

The data was obtained from Sydney University’s Instutional Analystics and Planning (IAP) department. It reflects the grades of students studying the top 15 junior Mathematics units in 2018 at the University of Sydney.

Each row represents one student per unit of study while each column represents their characteristics.

2.4 Assessing Stakeholders

Industry corporations: This report is of great use for industry corporations in targeting junior Sydney University students with high marks in mathematics. The discoveries made here are highly beneficial in attracting the most desirable junior interns or employees for math-focussed roles. Their specific characteristics have been well-explored throughout this research paper.

2.5 Initial Questions

1. How large is the data set?

# Looking at the dimensions of the data
cat(sprintf("Rows: %d\nColumns: %d", nrow(data), ncol(data)))

## Rows: 10845
## Columns: 23

The data set has 23 columns (variables) and 10 845 rows (entries).

2. What is the female-male split in junior Mathematics units?

# Finding the percent of females and males in the data set
percent_female = sum(data$gender == "F") * 100/nrow(data)
percent_male = sum(data$gender == "M") * 100/nrow(data)

# Printing result to the nearest %
cat(sprintf("Females = %.0f%%\nMales = %.0f%%", percent_female, percent_male))

## Females = 46%
## Males = 54%

From above, there are 8% more males than females who studied the top 15 junior Mathematics units in 2018.

There is a likelihood that this does not reflect the overall distribution of gender at the University of Sydney. In fact, the data set represents only a small sample of the total amount of students enrolled at the University of Sydney (59 129 in 2017).

3. How many students are enrolled in each unit of study?

# Creating a frequency table according to unit of study
data_uos = table(data$unit_of_study_identifier)
`Mathematics Units in 2018` = data$unit_of_study_identifier

# Plotting the table
tab1(`Mathematics Units in 2018`, sort.group = "decreasing", cum.percent = F, graph = F)

## Mathematics Units in 2018 : 
##         Frequency Percent
## UNIT S       1815    16.7
## UNIT O       1790    16.5
## UNIT H       1368    12.6
## UNIT V       1077     9.9
## UNIT Q        876     8.1
## UNIT G        645     5.9
## UNIT X        638     5.9
## UNIT K        614     5.7
## UNIT A        522     4.8
## UNIT C        287     2.6
## UNIT Y        258     2.4
## UNIT W        248     2.3
## UNIT T        240     2.2
## UNIT M        234     2.2
## UNIT Z        233     2.1
##   Total     10845   100.0

Thus, about 1800 students studied the most popular junior mathematics unit of study in 2018. With the least popular accommodating around 200 or 2% of the total.

# Looking at the mean number of students per unit
cat(sprintf("Mean students per unit: %d", mean(data_uos)))

## Mean students per unit: 723

While each unit of study accommodated 723 students on average.

The analysis above can also be visualised using circle packing:

# Creating frequency tables for each unit of study by semester
uos_sem_1 = data$unit_of_study_identifier[data$semester == 1]
data_uos_sem_1 = data.frame(table(uos_sem_1))

uos_sem_2 = data$unit_of_study_identifier[data$semester == 2]
data_uos_sem_2 = data.frame(table(uos_sem_2))

# Changing the class of `semester` to a factor variable
data_uos_sem_1$semester = as.factor(1)
data_uos_sem_2$semester = as.factor(2)

# Merging the frequency tables for semester 1 and 2 into one data set called `data_uos_sem`
data = merge(data, data_uos_sem_1, by.x = c("unit_of_study_identifier", "semester"), by.y = c("uos_sem_1", "semester"), all.x = T)
data = merge(data, data_uos_sem_2, by.x = c("unit_of_study_identifier", "semester"), by.y = c("uos_sem_2", "semester"), all.x = T)

# Combining frequency columns into a single column 
data$freq = with(data, pmax(Freq.x, Freq.y, na.rm = TRUE))

# Removing old columns
drops = c("Freq.x","Freq.y")
data = data[, !(names(data) %in% drops)]

# Creating a uniform vector of 1's
data$students = 1

# Creating `data_grouped` to summarise the number of students per group
data_grouped = data %>% 
  group_by(unit_of_study_identifier, semester, age_category, mode_of_study, gender, dom_int) %>%
  summarise(size = sum(students))

# Merging `data_grouped` with `data`
data = merge(data, data_grouped, all.x = T)

# Creating a phylogenetic path for each unit of study
data$pathString = paste("Top 15 Junior Maths Subjects", data$unit_of_study_identifier, data$semester, data$age_category, data$mode_of_study, data$gender, data$dom_int, sep = "/")

# Creating the 'tree_uos_sem' data tree structure
tree_uos_sem = as.Node(data)

# Circle packing of the phylogenetic path coloured by depth
circlepackeR(tree_uos_sem, color_min = "RGB(0,100,150)", color_max = "RGB(200,200,0)")

Similarly to the table above, this visualisation shows that the most enrolled units of study are UNIT S, O and H in descending order.

2.6 Limitations

Outliers have been excluded from the data set or altered to preserve anonymity. This includes students:

That didn’t record any access to Canvas over the whole semester
Who obtained non-standard grades (e.g. discontinuation, withdrawal, absent fail and failed requirements)
Who identify as neither male nor female, whom have instead been recorded in the data set as female

2.7 Ethical Concerns

No serious ethical concerns with distributing the data set or its results seem to be apparent.

This is mainly because the data has been completely anonymised with arbitrary UNIT _ names as well as broad age categories and the removal of distinct outliers.

However, one still may be able to discern information from the data set about a certain individual’s characteristics based on specific knowledge of their mark or tendency to access Canvas.

2.8 Domain Knowledge

Full time is defined as taking 18 or more credit points as a domestic student, or 24 as a student visa holder, in one semester.

Part time students have a reduced study load of less than 18 credit points per semester.

Performance bands for various marks ranges include:

Result Code	Result Name	Mark Range
HD	High Distinction	85 - 100
DI	Distinction	75 - 84
CR	Credit	65 - 74
PS	Pass	50 - 64
FA	Fail	0 - 49

There are 29 junior mathematics units of study at offer at the University of Sydney, including:

Advanced Units
Normal Units
Fundamental Units
Other Units (not available to all students)

Thus, only half (15/29) of the available units of study have been recorded in this data set.

3 Research Question

3.1 Who performs the best and worst at junior-level mathematics at the University of Sydney?

In this research question I have assumed that the current academic performance of the students studying junior mathematics at the University of Sydney has changed by little between 2018 and 2019. In fact, according to Times Higher Education the University of Sydney’s score for teaching has remained constant since 2017, while it’s ranking position has stayed around 60th since 2015. Thus, I have extrapolated that the findings from this data set serve as a general truth in the current year of 2019.

Comparing Mark Distributions

First of all, in order to begin we will need to create a data set with unique groupings for each type of student according to their characteristics. Namely, according to the qualitative variables gender, age, semester, mode of study and domestic or international status. For analysis purposes we should also find the median mark and IQR of each group, and record them as two columns in our matrix.

# Creating a new data frame `groups` with all possible combinations of the qualitative variables in `data`. Also aggregating to find each group's median and IQR.
grouped_median = aggregate(data$mark, list(data$gender, data$age_category, data$semester, data$mode_of_study, data$dom_int), median)
grouped_iqr = aggregate(data$mark, list(data$gender, data$age_category, data$semester, data$mode_of_study, data$dom_int), IQR)

# Merging the aggregates to form `groups`
groups = merge(grouped_median, grouped_iqr)

# Changing to appropriate column names
colnames(groups) = c("gender", "age_category", "semester", "mode_of_study", "dom_int", "median_mark", "iqr_mark")

Now, let’s write a function to compare two distributions and find which one occurs more often over larger values. This will equip us with the ability to compare the mark distributions of every group of maths students and thus find trends in the best and worst performing groups.

# Making a function to find the difference between two distributions
compare_dist = function(dist1_str, dist2_str) {
  
  # Evaluating the input strings as matrix paths
  dist1 = eval(parse(text = dist1_str))
  dist2 = eval(parse(text = dist2_str))
  
  # Accounting for distributions that have less than 2 elements
  if (length(dist1) < 2) {
    return(-1)
  }
  if (length(dist2) < 2) {
    return(0)
  }
  
  # Creating density objects for both distributions
  dens1 = density.default(dist1)
  dens2 = density.default(dist2)
  
  # Finding the average median of both distributions
  avg_median = mean(median(dist1), median(dist2))
  
  # Accounting for the average median being out of the range of either distribution
  if (avg_median > min(max(dens1$x), max(dens2$x))) {
    if (median(dist1) > median(dist2)) {
      return(1)
    }
    return(0)
  }
  if (avg_median < max(min(dens1$x), min(dens2$x))) {
    if (median(dist1) > median(dist2)) {
      return(1)
    }
    return(0)
  }
  
  # Returning the integral difference between the first and second density distributions. Integrating from the average median value to the maximum x value in the distribution's range.
  return(integrate.xy(dens1$x, dens1$y, a = avg_median) - integrate.xy(dens2$x, dens2$y, a = avg_median))
}

The above function compare_dist compares two distributions by measuring the difference in the area under their curves from the average of their median values to the maximum \(x\) values in their range. This idea is mathematically represented as follows:

\[ a = \frac{med(x_A) + med(x_B)}{2} \] \[ difference = \int_a^{max(x_A)} A(x)~dx - \int_a^{max(x_B)} B(x)~dx \]

Let’s write a wrapper function that applies compare_dist to a vector of distributions and returns an ordered vector of distribution ranks.

# Function to count how many distributions a certain distribution is greater than. Returns `success_vector` with this information.
loop_compare = function(vect) {
  
  # Creating an empty vector with entries 0
  success_vect = rep(0, length(vect))
  
  # Looping through each distribution
  for (i in 1:length(vect)) {
    successful = 0
    for (j in 1:length(vect)) {
      test = compare_dist(vect[i], vect[j])
      
      # Using 0.01 rather than 0 as a margin of error
      if (test >= 0.01) {
        successful = successful + 1
        success_vect[i] = successful
      }
      
      # Accounting for distributions of less than 2 elements
      if (test == -1) {
        success_vect[i] = -1
        break
      }
    }
  }
  return(success_vect)
}

Above, loop_compare returns the ordered vector success_vect with integer values representing the number of distributions that a certain distribution is greater than according to compare_dist. I use 0.01 or larger as my indicator that distribution A is more likely observes larger marks than distribution B. I believe that any difference equal to or greater than this is statistically significant.

Now we can apply this functionality to all the possible combinations of students in groups and thus find the best and worst performing groups of students studying junior mathematics.

# Assigning all possible distributions to `dist_vector`
dist_vector = apply(groups, 1, function(x) paste0("data$mark[data$gender == ", "'", x[1], "'", " & data$age_category == ", "'", x[2], "'", " & data$semester == ", x[3], " & data$mode_of_study == ", "'", x[4], "'", " & data$dom_int == ", "'", x[5], "'", "]"))

# Comparing all possible distributions
success = loop_compare(dist_vector)

# Binding the `success` vector to the `groups` data frame
groups = cbind(groups, success)

# Table of the best and worst 8 rows in the data set
dust(groups[with(groups, order(success, decreasing = T)),][c(1:8, 45:52),]) %>%
  sprinkle_border(8, border = "bottom")

gender	age_category	semester	mode_of_study	dom_int	median_mark	iqr_mark	success
F	Over 25	2	Full Time	International	74	5.75	51
F	18 and Under	2	Full Time	Domestic	72	19	50
M	18 and Under	2	Full Time	Domestic	72	18	49
F	18 and Under	1	Full Time	Domestic	71	17	47
M	18 and Under	1	Full Time	Domestic	71	18.75	46
F	18 and Under	2	Full Time	International	70.5	20.25	45
F	Over 25	1	Full Time	International	72.5	8.5	44
M	19-21	2	Full Time	Domestic	70	17.25	42
M	19-21	1	Part Time	Domestic	57	13	7
F	19-21	1	Part Time	Domestic	54.5	33	6
F	22-25	1	Part Time	International	52	7.5	6
F	18 and Under	1	Part Time	Domestic	54	4	5
M	22-25	1	Part Time	Domestic	51	30.25	3
M	18 and Under	2	Part Time	Domestic	40.5	32	1
M	22-25	2	Part Time	International	42	1	1
F	19-21	1	Part Time	International	40.5	9.5	0

The split table above shows 8 groups of students who performed the best in junior mathematics during 2018, followed by 8 who performed the worst, both in descending order. This judgement of mark performance is made according to the success parameter, which measures how many groups of students a certain group attained larger marks than.

Some what surprisingly, the best performing group was full time international women over 25. However this only represented 6 individuals in 2018, a small population which may not be entirely reflective of their year-to-year performance.

Nonetheless, full time domestic male and female students, aged 18 and below, performed almost exactly the same in both semesters with common median marks of 71 and 72 in semesters 1 and 2 respectively. This marginal difference between semesters may be due to younger students adjusting to university life over the course of the year. In spite of these commonalities however, according to our distribution comparison tests females out-performed males in both semesters for this category of students.

Overall, it appears that the best performing groups were:

Full time domestic students 18 and under, as well as
Full time international women

On the other hand, the worst performing group was part time international women aged 19-21 in semester 1. Note that international women appear in both extremes of the mark spectrum.

Only marginally better in performance were part-time international men aged 22-25 and part time domestic men aged 18 and under, both in semester 2. Interestingly, they seemed to perform much worse in semester 2 as opposed to the beginning of the year, perhaps suggesting that less inclined students lose motivation over the course of the year. Their median mark was 42 and 40.5 respectively, receiving a fail on average.

Overall, the worse performing groups appear to be characterised by:

Part time domestic students in semester 1

In order to visualise our results let’s compare the mark distribution for the best and worst performing groups of students with a comparative density plot:

# Binding `dist_vector` the `groups` data frame
groups = cbind(groups, dist_vector)

# Subsetting `groups` into the best and worst performing 8 groups
groups_top8 = groups %>%
  top_n(8, success) %>%
  arrange(desc(success)) %>%
  mutate(performance = "Best 8")

groups_bottom8 = groups %>%
  top_n(-12, success) %>%
  top_n(8, success) %>%
  arrange(desc(success)) %>%
  mutate(performance = "Worse 8")

groups_subset = rbind(groups_top8, groups_bottom8)

# Adding a legend string for each group
groups_subset$legend = paste(groups_subset$mode_of_study, ifelse(groups_subset$gender == "M", "Male", "Female"), groups_subset$dom_int, "students", groups_subset$age_category, "in", ifelse(groups_subset$semester == 1, "Semester 1", "Semester 2"), sep = " ")

# Merging `groups_subset` with `data` to form `data_subset`
data_subset = merge(data, groups_subset)

# Density plot of the mark distribution for the best and worst 8 groups of students
ggplotly(ggplot(data_subset, aes(mark, fill = legend, success = success)) +
           geom_density(aes(color = legend), alpha = 0.8, position = "stack") +  
           facet_wrap(~performance, nr = 2) +
           ggtitle("Mark Density Distribution for the Best and Worst 8 Groups of Students") +
           theme_bw())

The density plot of the mark distribution for each group of students above clearly indicates a juxtaposition between the stacked distribution for the best and worst 8 groups. In particular, the median appears to closer to 75 for the top performing groups, in comparison to 50 for those in the bottom tier.

Interestingly, there appears to be a dependency between the mode of study and mark distribution of each group of students. In particular, part time individuals seem to consistently score lower marks than those who study full time. This may be because full time students are more dedicated to their studies and pressured to keep up to date with their work.

Characterising High Distinctions and Failure

Now, let’s examine the characteristics of high distinction, distinction and failing students specifically. This knowledge should be of great interest to the stakeholder for targeting intelligent junior mathematicians from the University of Sydney.

Let’s explore this by looking at the percentage difference between HD, DI or FA awarded students and the whole cohort for each characteristic. Note that the HD (High Distinction), DI (Distinction) and FA (Fail) awards correspond to these mark intervals.

# Subsetting `data` for students who received a HD into `data_hd`, a DI into `data_di` and an FA into `data_fa`
data_hd = data[data$mark >= 85 & data$mark <= 100 & data$mode_of_study != "Unknown",]
data_di = data[data$mark >= 75 & data$mark < 85 & data$mode_of_study != "Unknown",]
data_fa = data[data$mark >= 0 & data$mark < 50 & data$mode_of_study != "Unknown",]

# Creating a new data set `data_clean` without "Unknown" as a `mode_of_study` since it obscures analysis
data_clean = data %>%
  mutate(mode_of_study = str_remove(mode_of_study, "Unknown"))

# Finding the proportions of all students in each categorical variable
percent_sem = as.matrix(prop.table(table(data_clean$semester)) * 100)
percent_study = as.matrix(prop.table(table(data_clean$mode_of_study)) * 100)
percent_gender = as.matrix(prop.table(table(data_clean$gender)) * 100)
percent_age = as.matrix(prop.table(table(data_clean$age_category)) * 100)
percent_domint = as.matrix(prop.table(table(data_clean$dom_int)) * 100)

# Removing a rogue row
percent_study = percent_study[-1,]

# Creating proportion tables for HD, DI and FA students in each categorical variable of interest
percent_hd_sem = as.matrix(prop.table(table(data_hd$semester)) * 100)
percent_hd_study = as.matrix(prop.table(table(data_hd$mode_of_study)) * 100)
percent_hd_gender = as.matrix(prop.table(table(data_hd$gender)) * 100)
percent_hd_age = as.matrix(prop.table(table(data_hd$age_category)) * 100)
percent_hd_domint = as.matrix(prop.table(table(data_hd$dom_int)) * 100)

percent_di_sem = as.matrix(prop.table(table(data_di$semester)) * 100)
percent_di_study = as.matrix(prop.table(table(data_di$mode_of_study)) * 100)
percent_di_gender = as.matrix(prop.table(table(data_di$gender)) * 100)
percent_di_age = as.matrix(prop.table(table(data_di$age_category)) * 100)
percent_di_domint = as.matrix(prop.table(table(data_di$dom_int)) * 100)

percent_fa_sem = as.matrix(prop.table(table(data_fa$semester)) * 100)
percent_fa_study = as.matrix(prop.table(table(data_fa$mode_of_study)) * 100)
percent_fa_gender = as.matrix(prop.table(table(data_fa$gender)) * 100)
percent_fa_age = as.matrix(prop.table(table(data_fa$age_category)) * 100)
percent_fa_domint = as.matrix(prop.table(table(data_fa$dom_int)) * 100)

# Removing "Unknown" as a mode_of_study since it obscures analysis
percent_hd_study = percent_hd_study[-3,]
percent_di_study = percent_di_study[-3,]
percent_fa_study = percent_fa_study[-3,]

# Calculating the difference between HD, DI and FA students and the entire cohort
hd_sem_diff = as.data.frame(percent_hd_sem - percent_sem)
hd_study_diff = as.data.frame(percent_hd_study - percent_study)
hd_gender_diff = as.data.frame(percent_hd_gender - percent_gender)
hd_age_diff = as.data.frame(percent_hd_age - percent_age)
hd_domint_diff = as.data.frame(percent_hd_domint - percent_domint)

di_sem_diff = as.data.frame(percent_di_sem - percent_sem)
di_study_diff = as.data.frame(percent_di_study - percent_study)
di_gender_diff = as.data.frame(percent_di_gender - percent_gender)
di_age_diff = as.data.frame(percent_di_age - percent_age)
di_domint_diff = as.data.frame(percent_di_domint - percent_domint)

fa_sem_diff = as.data.frame(percent_fa_sem - percent_sem)
fa_study_diff = as.data.frame(percent_fa_study - percent_study)
fa_gender_diff = as.data.frame(percent_fa_gender - percent_gender)
fa_age_diff = as.data.frame(percent_fa_age - percent_age)
fa_domint_diff = as.data.frame(percent_fa_domint - percent_domint)

# Changing to appropriate column names
colnames(hd_sem_diff)[1] = "HD"
colnames(hd_study_diff)[1] = "HD"
colnames(hd_gender_diff)[1] = "HD"
colnames(hd_age_diff)[1] = "HD"
colnames(hd_domint_diff)[1] = "HD"

colnames(di_sem_diff)[1] = "DI"
colnames(di_study_diff)[1] = "DI"
colnames(di_gender_diff)[1] = "DI"
colnames(di_age_diff)[1] = "DI"
colnames(di_domint_diff)[1] = "DI"

colnames(fa_sem_diff)[1] = "FA"
colnames(fa_study_diff)[1] = "FA"
colnames(fa_gender_diff)[1] = "FA"
colnames(fa_age_diff)[1] = "FA"
colnames(fa_domint_diff)[1] = "FA"

# Joining HD, DI and FA students
sem_diff = cbind(hd_sem_diff, di_sem_diff, fa_sem_diff)
study_diff = cbind(hd_study_diff, di_study_diff, fa_study_diff)
gender_diff = cbind(hd_gender_diff, di_gender_diff, fa_gender_diff)
age_diff = cbind(hd_age_diff, di_age_diff, fa_age_diff)
domint_diff = cbind(hd_domint_diff, di_domint_diff, fa_domint_diff)

# Customising hover text
sem_diff$hover_text_hd = sprintf("Semester %s | %.2f%%", row.names(sem_diff), sem_diff$HD)
sem_diff$hover_text_di = sprintf("Semester %s | %.2f%%", row.names(sem_diff), sem_diff$DI)
sem_diff$hover_text_fa = sprintf("Semester %s | %.2f%%", row.names(sem_diff), sem_diff$FA)

study_diff$hover_text_hd = sprintf("%s | %.2f%%", row.names(study_diff), study_diff$HD)
study_diff$hover_text_di = sprintf("%s | %.2f%%", row.names(study_diff), study_diff$DI)
study_diff$hover_text_fa = sprintf("%s | %.2f%%", row.names(study_diff), study_diff$FA)

gender_diff$hover_text_hd = sprintf("%s | %.2f%%", ifelse(row.names(gender_diff) == "F", "Female", "Male"), gender_diff$HD)
gender_diff$hover_text_di = sprintf("%s | %.2f%%", ifelse(row.names(gender_diff) == "F", "Female", "Male"), gender_diff$DI)
gender_diff$hover_text_fa = sprintf("%s | %.2f%%", ifelse(row.names(gender_diff) == "F", "Female", "Male"), gender_diff$FA)

age_diff$hover_text_hd = sprintf("%s | %.2f%%", row.names(age_diff), age_diff$HD)
age_diff$hover_text_di = sprintf("%s | %.2f%%", row.names(age_diff), age_diff$DI)
age_diff$hover_text_fa = sprintf("%s | %.2f%%", row.names(age_diff), age_diff$FA)

domint_diff$hover_text_hd = sprintf("%s | %.2f%%", row.names(domint_diff), domint_diff$HD)
domint_diff$hover_text_di = sprintf("%s | %.2f%%", row.names(domint_diff), domint_diff$DI)
domint_diff$hover_text_fa = sprintf("%s | %.2f%%", row.names(domint_diff), domint_diff$FA)

# Bar plot of these percentage changes
bar_sem = plot_ly(sem_diff, x = ~row.names(sem_diff), y = ~HD, type = "bar", name = "HD", text = ~hover_text_hd, hoverinfo = "text", marker = list(color = 'rgba(50, 171, 96, 0.7)', line = list(color = 'rgba(50, 171, 96, 1.0)', width = 1.5))) %>%
  add_trace(y = ~DI, name = "DI", text = ~hover_text_di, marker = list(color = 'rgba(55, 128, 191, 0.7)', line = list(color = 'rgba(55, 128, 191, 1.0)', width = 1.5))) %>%
  add_trace(y = ~FA, name = "FA", text = ~hover_text_fa, marker = list(color = 'RGBA(0,0,0,0.35)', line = list(color = 'RGBA(0,0,0,0.45)', width = 1.5))) %>%
  layout(title = "Percentage Difference Between HD, DI and FA Students and the Whole Cohort in Each Category", yaxis = list(title = "Percentage Change (%)", range = c(-10, 10)), legend = list(x = 0, y = 0.95, bgcolor = "#E2E2E2", bordercolor = "#FFFFFF", borderwidth = 2), paper_bgcolor = 'rgba(245, 246, 249, 1)', plot_bgcolor = 'rgba(245, 246, 249, 1)')

bar_study = plot_ly(study_diff, x = ~row.names(study_diff), y = ~HD, type = "bar", name = "HD", text = ~hover_text_hd, hoverinfo = "text", marker = list(color = 'rgba(50, 171, 96, 0.7)', line = list(color = 'rgba(50, 171, 96, 1.0)', width = 1.5)), showlegend = F) %>%
  add_trace(y = ~DI, name = "DI", text = ~hover_text_di, marker = list(color = 'rgba(55, 128, 191, 0.7)', line = list(color = 'rgba(55, 128, 191, 1.0)', width = 1.5))) %>%
  add_trace(y = ~FA, name = "FA", text = ~hover_text_fa, marker = list(color = 'RGBA(0,0,0,0.35)', line = list(color = 'RGBA(0,0,0,0.45)', width = 1.5))) %>%
  layout(yaxis = list(title = "", range = c(-10, 10), showticklabels = F))

bar_gender = plot_ly(gender_diff, x = ~row.names(gender_diff), y = ~HD, type = "bar", name = "HD", text = ~hover_text_hd, hoverinfo = "text", marker = list(color = 'rgba(50, 171, 96, 0.7)', line = list(color = 'rgba(50, 171, 96, 1.0)', width = 1.5)), showlegend = F) %>%
  add_trace(y = ~DI, name = "DI", text = ~hover_text_di, marker = list(color = 'rgba(55, 128, 191, 0.7)', line = list(color = 'rgba(55, 128, 191, 1.0)', width = 1.5))) %>%
  add_trace(y = ~FA, name = "FA", text = ~hover_text_fa, marker = list(color = 'RGBA(0,0,0,0.35)', line = list(color = 'RGBA(0,0,0,0.45)', width = 1.5))) %>%
  layout(yaxis = list(title = "", range = c(-10, 10), showticklabels = F))

bar_age = plot_ly(age_diff, x = ~row.names(age_diff), y = ~HD, type = "bar", name = "HD", text = ~hover_text_hd, hoverinfo = "text", marker = list(color = 'rgba(50, 171, 96, 0.7)', line = list(color = 'rgba(50, 171, 96, 1.0)', width = 1.5)), showlegend = F) %>%
  add_trace(y = ~DI, name = "DI", text = ~hover_text_di, marker = list(color = 'rgba(55, 128, 191, 0.7)', line = list(color = 'rgba(55, 128, 191, 1.0)', width = 1.5))) %>%
  add_trace(y = ~FA, name = "FA", text = ~hover_text_fa, marker = list(color = 'RGBA(0,0,0,0.35)', line = list(color = 'RGBA(0,0,0,0.45)', width = 1.5))) %>%
  layout(yaxis = list(title = "", range = c(-10, 10), showticklabels = F))

bar_domint = plot_ly(domint_diff, x = ~row.names(domint_diff), y = ~HD, type = "bar", name = "HD", text = ~hover_text_hd, hoverinfo = "text", marker = list(color = 'rgba(50, 171, 96, 0.7)', line = list(color = 'rgba(50, 171, 96, 1.0)', width = 1.5)), showlegend = F) %>%
  add_trace(y = ~DI, name = "DI", text = ~hover_text_di, marker = list(color = 'rgba(55, 128, 191, 0.7)', line = list(color = 'rgba(55, 128, 191, 1.0)', width = 1.5))) %>%
  add_trace(y = ~FA, name = "FA", text = ~hover_text_fa, marker = list(color = 'RGBA(0,0,0,0.35)', line = list(color = 'RGBA(0,0,0,0.45)', width = 1.5))) %>%
  layout(yaxis = list(title = "", range = c(-10, 10), showticklabels = F))

subplot(bar_sem, bar_study, bar_gender, bar_age, bar_domint, titleY = T)

As you can see in the comparative bar plot above, there are unique differences between HD, DI and FA awarded students and the general cohort of junior mathematicians.

Students were more likely to achieve a High Distinction if they were:

In semester 1
Studying full time
Male
Aged 18 and under
and Domestic

Whereas, Distinction awarded students were generally:

Not found more than usual in a particular semester
Studying full time
Female
Aged 18 and under
Only marginally (1.71%) more domestic than the overall cohort

In contrast, students who obtained a Fail were more than usual:

Not found in a particular semester
Studying part time
Male
Aged over 25
International

Since there appears to be no clear distinction between high and low performing students in their semester of study or gender, we can disregard a dependence relationship between mark achieved and these categorical variables. In fact, according to the 2014 journal article “The Science of Sex Differences in Science and Mathematics” there tends to be “more males at both high- and low-ability extremes”. This is evident in the results shown above, wherein males are more likely to be awarded a High Distinction or Fail than the average cohort.

However, three particular characteristics of higher performing students stand out. Namely they are generally:

Enrolled full time
Aged 18 and under
Domestic in origin

Interestingly, according to a media article published in 2019 “international students overall fail a larger share of the subjects they take than domestic students”. They attribute these results to growing concerns regarding the admission of international students to Australian universities without the necessary English-language skills.

The aforementioned conclusions are momentous in targeting high performing intelligent junior mathematicians from the University of Sydney for internships or employment in math-focussed roles.

Moreover, we have solved in particular our research question about who performs the best and worst at junior-level mathematics at the University of Sydney.

4 References

Annual Report 2017 (Rep.). (2018, April). Retrieved May 4, 2019, from The University of Sydney website: https://sydney.edu.au/content/dam/corporate/documents/about-us/values-and-visions/Annual-report-2.pdf#page=12

University of Sydney Breakdown. (2019, May 08). Retrieved May 22, 2019, from https://www.timeshighereducation.com/world-university-rankings/university-sydney#survey-answer

Guide to grades. (n.d.). Retrieved May 22, 2019, from https://sydney.edu.au/students/guide-to-grades.html

Change your study load. (n.d.). Retrieved May 24, 2019, from https://sydney.edu.au/students/change-study-load.html

Junior Mathematics and Statistics. (n.d.). Retrieved May 24, 2019, from https://www.maths.usyd.edu.au/u/UG/JM/

Halpern, D. F., Benbow, C. P., Geary, D. C., Gur, R. C., Hyde, J. S., & Gernsbacher, M. A. (2007). The Science of Sex Differences in Science and Mathematics. Psychological science in the public interest : a journal of the American Psychological Society, 8(1), 1-51. doi:10.1111/j.1529-1006.2007.00032.x

Norton, A. (2019, May 10). Are international students passing university courses at the same rate as domestic students? Retrieved May 25, 2019, from http://theconversation.com/are-international-students-passing-university-courses-at-the-same-rate-as-domestic-students-116666

5 Session Info

sessionInfo()

## R version 3.5.2 (2018-12-20)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 17134)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_Australia.1252  LC_CTYPE=English_Australia.1252   
## [3] LC_MONETARY=English_Australia.1252 LC_NUMERIC=C                      
## [5] LC_TIME=English_Australia.1252    
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] sfsmisc_1.1-4           pixiedust_0.8.6        
##  [3] circlepackeR_0.0.0.9000 data.tree_0.7.8        
##  [5] forcats_0.4.0           stringr_1.4.0          
##  [7] dplyr_0.8.0.1           purrr_0.3.2            
##  [9] readr_1.3.1             tidyr_0.8.3            
## [11] tibble_2.1.1            tidyverse_1.2.1        
## [13] plotly_4.9.0            ggplot2_3.1.1          
## [15] prettydoc_0.2.1         epiDisplay_3.5.0.1     
## [17] nnet_7.3-12             MASS_7.3-51.1          
## [19] survival_2.43-3         foreign_0.8-71         
## [21] DT_0.5                  janitor_1.1.1          
## [23] kableExtra_1.1.0        knitr_1.22.8           
## 
## loaded via a namespace (and not attached):
##  [1] nlme_3.1-137       lubridate_1.7.4    webshot_0.5.1     
##  [4] RColorBrewer_1.1-2 httr_1.4.0         tools_3.5.2       
##  [7] backports_1.1.4    R6_2.4.0           lazyeval_0.2.2    
## [10] colorspace_1.4-1   withr_2.1.2        tidyselect_0.2.5  
## [13] gridExtra_2.3      compiler_3.5.2     cli_1.1.0         
## [16] rvest_0.3.3        xml2_1.2.0         influenceR_0.1.0  
## [19] labeling_0.3       checkmate_1.9.1    scales_1.0.0      
## [22] digest_0.6.18      rmarkdown_1.12     pkgconfig_2.0.2   
## [25] htmltools_0.3.6    htmlwidgets_1.3    rlang_0.3.4       
## [28] readxl_1.3.1       rstudioapi_0.10    shiny_1.3.1       
## [31] visNetwork_2.0.6   generics_0.0.2     jsonlite_1.6      
## [34] crosstalk_1.0.0    rgexf_0.15.3       magrittr_1.5      
## [37] Matrix_1.2-15      Rcpp_1.0.1         munsell_0.5.0     
## [40] viridis_0.5.1      stringi_1.4.3      yaml_2.2.0        
## [43] snakecase_0.9.2    plyr_1.8.4         grid_3.5.2        
## [46] promises_1.0.1     crayon_1.3.4       lattice_0.20-38   
## [49] haven_2.1.0        splines_3.5.2      hms_0.4.2         
## [52] pillar_1.3.1       igraph_1.2.4       XML_3.98-1.19     
## [55] glue_1.3.1         evaluate_0.13      downloader_0.4    
## [58] data.table_1.12.2  modelr_0.1.4       httpuv_1.5.1      
## [61] cellranger_1.1.0   gtable_0.3.0       assertthat_0.2.1  
## [64] xfun_0.6           mime_0.6           xtable_1.8-3      
## [67] broom_0.5.2        later_0.8.0        viridisLite_0.3.0 
## [70] Rook_1.1-1         DiagrammeR_1.0.0   brew_1.0-6

Characteristics of the Best and Worst Junior Mathematicians

Project 3 - Monday 9-11am Carslaw 353

SID: 490390494

The University of Sydney | DATA1901 | May 2019